Gareth Jones on number theory, why there are infinitely many of prime numbers and how to calculate the proportion of prime numbers up to any given n
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'Let's take the finite set of prime numbers and multiply them all together and call that integer n. So all the prime numbers we chose divide n. Now look at n+1, apply the fundamental theorem of arithmetic to that. It's a product of prime numbers, so there must be some prime number which divides it. That prime number cannot be one of the finite set we started with, because if it was, it would divide n and it would divide n+1, so it would have to divide their difference which is 1, but one is not divisible by any prime number, so we have produced a new prime number.'
Gareth Jones, Emeritus Professor of Mathematics, University of Southampton
Logic and Mathematics: http://serious-science.org/logic-and-mathematics-7243
Logical Paradoxes: http://serious-science.org/logical-paradoxes-7406
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